with \(\sigma _k\)  the Pauli matrices. So \(h(p)^*=h(p)\) , \(h(p)^2=\epsilon (p)^2I_4\) 
with \(\epsilon (p)=\sqrt{|p|^2+m^2}\) . As to \(q^\eta \) , recall \(\mathfrak {p}^{\eta }=(\eta \epsilon (p),p)\)  and \(\mathfrak {q}^\eta = (q_0^\eta ,q^\eta )=A^{-1}\cdot \mathfrak {p}^\eta \)  from Eq. (REF ).
Furthermore, \(\pi ^\eta (p):=\frac{1}{2}\big (I_2+\frac{\eta }{\epsilon (p)}h(p)\big )\)  for \(\eta \in \lbrace +,-\rbrace \) . The multiplication operator
by \(\pi ^\eta \)  is the orthogonal projection onto the subspace of positive energy
(\(\eta =+\) ) and negative energy (\(\eta =-\) ), respectively.
Finally, in the Weyl representation one has \(s(A):=\operatorname{diag}(A,A^{* -1})\) .Let \(S\in U(4)\)  and let \(\text{\small {{\it S}}}\)  be the unitary operator on \(L^2(\mathbb {R}^3,4)\)  with \((\text{\small {{\it S}}}\psi )(x):=S\,\psi (x)\) . Then
\(\text{\small {{\it S}}}\;W^{\textsc {d}\,mom}(g)\,\text{\small {{\it S}}}^{-1}\)  is given by the right hand side of Eq. (REF ) with respect to
\(\beta ^{\prime }:= S \,\beta \,S^{-1}\) , \(\alpha ^{\prime }_k:=S\,\alpha _k\,S^{-1}\) , and \(s^{\prime }(A):=S\,s(A)\,S^{-1}\) . — Moreover, there is a formula valid for any permissible choice (cf. [1]}) of the \(\beta \) , \(\alpha \) 's, cf. [2]}
\(s(A)=\big (a_01_4+\sum _{k=1}^3a_k\gamma _5 \alpha _k\big )\frac{1}{2}(1_4+\gamma _5)+\big (\bar{a}_01_4+\sum _{k=1}^3\bar{a}_k\gamma _5 \alpha _k\big )\frac{1}{2}(1_4-\gamma _5)\) 
where \(\gamma _5:=-i\alpha _1\alpha _2\alpha _3\)  and \(A=a_01_2+\sum _{k=1}^3a_k\sigma _k\)  with \(a_0,a_k\in , \) a20-a12-a22-a32=1\(.\)

